2 When is a 2-Digit Number the Sum of the Squares of its Digits?
|
|
- Brice Bates
- 7 years ago
- Views:
Transcription
1 When Does a Number Equal the Sum of the Squares or Cubes of its Digits? An Exposition and a Call for a More elegant Proof 1 Introduction We will look at theorems of the following form: by William Gasarch Theorem 1.1 Let L < U and k be in N. For every L x U there are XXX numbers that are the sum of the kth powers of their digits. The theorems we prove are easy and known. We will give several proofs of some of them with an eye towards the question: What is an elegant proof? For the last theorem I have a proof that is not elegant. If you know a more elegant one then please let me know. For each proof we note how many times we had to actually CHECK a number. This is a crude measure of elegance the fewer checks the more elegant the proof (compared to other proofs of the same theorem). We try not to game the system by having rather elaborate proofs that don t check that much. 2 When is a 2-Digit Number the Sum of the Squares of its Digits? We will prove the following theorem several different ways. Theorem 2.1 If 11 x 99 then x is not the sum of the squares of its digits. 2.1 Proof by Enumeration One can prove this theorem by listing all numbers between 11 and 99 and seeing that none of them work. One could write a program to produce this proof. We omit this proof. Number of Checks: : 89. PROS: You get the answer easily. If you wrote a program then you can adjust it to find the answers. to many more problems of this type (e.g., How many 3-digits numbers are the sum of the cubes of their digits) or variants of this problem (e.g., How many 2-digit numbers are within 1 of the sum of their digits? Which ones are they?) CONS: You get no insights. 2.2 Proof by Looking at each 10-number interval Case 1: x = 10a where 1 a 9 (so x ends in a 0). a 2 = 10a a = 10. 1
2 So this cannot happen. Note we did no checks. Case 2: 11 x 19. Since 11, 12, 13 > > > , none of 11,12,13 work. Since 15, 16, 17, 18, 19 < < < < < , none of 15,16,17,18,19 work. The only number left is 14 which does not work since Note we did one check. Case 3: Look at 21 x , 22, 23, 24 > , 27, 28, 29 < does not work since Note that we did one check. Case 4: 31 x , 32, 33, 34 > , 37, 38, 39 < Note that we did zero checks. Case 5: 41 x , 42, 43, 44 > , 47, 48, 49 < Note that we did one check. Case 6: 51 x , 52, 53, 54, 55 > , 57, 58, 59 < Note that we did zero checks. Case 7: 61 x , 62, 63, 64 > , 67, 68, 69 < Note that we did one check. Case 8: 71 x , 72, 73, 74 > , 77, 78, 78 < Note that we did one check. Case 9: 81 x , 82, 83, 84 > , 87, 88, 89 < Note that we did one check. Case 9: 91 x , 92 > , 96, 97, 98, 99 < , Note that we did two checks. Number of Checks: 8 PRO: Only ten cases. The techniques used may be helpful for other problems. CON: Only ten cases? If you do other problems with way the number of cases may get rather large. 2.3 Proof Using Mod 2, Mod 4, and Mod 10 Assume, by way of contradiction, that there exists 1 a 9 and 0 b 9 such that 10a + b = a 2 + b 2 henceforth the main equation. Take this mod 2 using x 2 x (mod 2) and 10 0 (mod 2) to obtain a 0 (mod 2). 2
3 If we take the main equation mod 4, using a 2 0 (mod 4), we get Hence b {0, 1, 4, 5, 8, 9}. We now take the main equation mod 10. a = b(b 1) 0 (mod 4). b a 2 + b 2 (mod 10). b(b 1) (mod 10). We now run through b {0, 1, 4, 5, 8, 9} and note for which ones a = b(b 1) (mod 10) exists and is a nonzero even number. Note that the squares mod 10 are 0, 1, 4, 5, 6, 9 so the only nonzero even ones are 4, 6. b b(b 1) (mod 10) a = b(b 1) (mod 10) comment NO GOOD: Need NO GOOD: Need NONE NO GOOD: No Square Root NO GOOD: Need NO GOOD: Need , 8 OH, Will need to check 9 8 NONE NO GOOD: No Square Root So the only candidates for (a, b) are (2, 4), (8, 4). One can check that these do not satisfy the main equation. Number of Checks: 2 PRO: The technique may extends to other problems. CAVEAT: Is this proof really more elegant then the intervals proof? 3 When is a d-digit Number the Sum of the Squares of its Digits? Theorem 3.1 If x 2 then x is not the sum of the squares of its digits. Case 1: 2 x 9. Since x < x 2 the theorem is true for 2 x 9. Case 2: 10 x 99. By Theorem 2.1 our theorem is true for these x. Case 3: 1 a 9, 0 b 9, 0 c 9. Assume x = 100a + 10b + c = a 2 + b 2 + c 2. The largest x can be is < 300. Hence a {1, 2}. The largest x can be is = 176 < 200. Hence a = 1 and b 7. The largest x can be is = 141. Hence and b 4. The largest x can be is = 98 <
4 Hence there is no such x. Case 4: x Let x = b m b m 1 b 0 where 1 b m 9, for 0 i m 1, 0 b i 9, and m 3. Since b m 1 we have x 10 m. Since x is the sum of the squares of its digits, x 81(m+1) Summing up 10 m+1 x 81(m + 1). We leave it to the reader to show that if m 3 then this is impossible. Number of Checks: 2 for Case 2, but 0 for the rest. So just 2. 4 When is a 2-digit Number the Sum of the Cubes of its Digits Theorem 4.1 If 11 x 99 then x is not the sum of the cubes of its digits. Assume, by way of contradiction, that there exists 1 a 9, 0 b 9 such that 10a + b = a 3 + b 3 henceforth the main equation. Hence 10a + 9 a 3 so a 3 10a 9 0. We leave it to the reader to show that this implies a 3. Take the main equation mod 2, using a 3 a (mod 2), b 3 b (mod 2), 10 0 (mod 2) to obtain Since 1 a 3 and a 0 we get a 0 (mod 2). (mod 2), we have a = 2. Plugging a = 2 into the main equation b(b 1)(b + 1) = 12 Alas, 12 cannot be written as the product of three consecutive numbers. Contradiction. Number of Checks: 0. 5 When is a 3-digit Number the Sum of the Cubes of its Digits Theorem 5.1 There is exactly one number 100 x 999 that is the sum of the cubes of its digits. That number is 153. Assume that there exists 1 a 9, 0 b 9, 0 c 9 such that 100a + 10b + c = a 3 + b 3 + c 3 henceforth the main equation. We will obtain conditions on a, b, c that force a = 1, b = 5, c = 3. 4
5 Claim 1: a b (mod 2). Proof of Claim 1 Take the main equation mod 2 to get a + b + c c (mod 2). Hence a b (mod 2). End of Proof of Claim 1 Claim 2: 1. b, c 7 2. If b 4 then c If c 7 then b Assume 3 a 8. If b 6 then c 5. Proof of Claim 2 Note that b 3 + c 3 10b c = 100a a 3. We will need the following table. a 100a a By the table if 1 a 9 then 99 b 3 + c 3 10b c 384. By the table if 3 a 8 then 273 b 3 + c 3 10b c ) If b 8 or c 8 then clearly b 3 + c 3 10b c 385. Hence b, c 7. 2) If b 4 then b 3 10b 24 hence 99 c 3 c Therefore c 3 c 75. Hence c 5. 3) If c 7 and b 5 then b 3 + c 3 10b c > 384. Hence if c 7 then b 4. 4) (3 a 8) If b 6 then b 3 10b 156 hence 273 c 3 c Therefore c 3 c 117. Hence c 5. End of Proof of Claim 2 Claim 3: 1. If a b (mod 4) then c / {2, 6}. 2. If a b (mod 4) and a is odd then c {2, 6}. Proof of Claim 3: 1) We take the main equation mod 4 using that a 3 b 3 (mod 4). 2b + c 2b 3 + c 3 (mod 4). 5
6 Since one of b, 1 b, 1 + b is even we have c 3 c 2b 2b 3 2b(1 b)(1 + b) (mod 4). c 3 c 0 (mod 4). This is not satisfied by any c {2, 6}. 1) We take the main equation mod 4 using that a 3 + b 3 0 (mod 4). 2b + c c 3 (mod 4). 2b c 3 c (mod 4). Since a is odd, b is odd. The set of c that satisfy this is {2, 6}. End of Proof of Claim 3 Claim 4: If a 0 (mod 2) then there is no solution to the main equation. Proof of Claim 4: If a 0 (mod 2) then, by Claim 1, b 0 (mod 2). The main equation mod 8, using that a 3 b (mod 8) and 10 2 (mod 8), is 2b c 3 c (mod 8). We now present a table of c 3 c (mod 8) as 0 c 7, and the possible values of b. c c 3 c (mod 8) b 0 0 0, , , , 4 For 0 c 6 we need, by Claim 2, b 6. Hence none of these values for c work. The only case left is c = 7. If c = 7 then 100a + 10b + 7 = a 3 + b a + 10b = a 3 + b Since 100a + 10b 336 we have a 3. Since a is even a 4. 6
7 For this item all mods are mod 10. If we take the equation mod 10 we get 100a + 10b = a 3 + b a 3 + b 3 4. a a 3 (mod 10) b 3 = 4 a 3 (mod 10) b The a = 8 case has c = 7 and b = 8 which cannot happen by Claim 3. candidates to check are (4, 0, 7), (6, 0, 7). None of them work. Hence the only Note: This claim took 2 checks. End of Proof of Claim 4 We can now enumerate all of the possibilities left. By Claim 4 we need only look at a 1 (mod 2). By Claims 1 and 2 we can assume b {1, 3, 5, 7} and c a = 1: By Claim 2 if b 4 then c 5. Since 6 3, we have c 5. Since we have that (a, b, c) (1, 5, 5). Hence the possible (a, b, c) are (1, 1, 5), (1, 3, 5), (1, 5, 0), (1, 5, 1), (1, 5, 2), (1, 5, 3), (1, 5, 4). By Claim 3 (1, 3, 5) and (1, 5, 2) is eliminated. Hence the only ones left to check are Only (1, 5, 3) works. Note: This took 5 checks. (1, 1, 5), (1, 5, 0), (1, 5, 1), (1, 5, 3), (1, 5, 4). 2. a = 3. By Claim 2 if b 5 then c 6. Since a (a, b, c) / {(3, 1, 6), (3, 3, 6)}. Since a (a, b, c) (3, 6, 6). Since a (a, b, c) / {(3, 5, 7), (3, 7, 4), (3, 7, 5), (3, 7, 6), (3, 7, 7)}. Hence the possible (a, b, c) are {(3, 1, 7), (3, 3, 7), (3, 5, 6), (3, 7, 0), (3, 7, 1), (3, 7, 2), (3, 7, 3)}. By Claim 3 (3, 1, 7) and (3, 7, 2) are eliminated. Hence the only ones left to check are 7
8 {(3, 3, 7), (3, 5, 6), (3, 7, 0), (3, 7, 1), (3, 7, 3)}. None of these work. This took 5 checks. 3. a = 5. By Claim 2 if b 5 then c 6. Since a (a, b, c) / {(5, 1, 6), (5, 3, 6), (5, 5, 6)}. Since a (a, b, c) / {(5, 7, 3), (5, 7, 4), (5, 7, 5), (5, 7, 7), (5, 7, 6), (5, 7, 7), (5, 3, 7), (5, 5, 7)}. The only cases left to check is (a, b, c) = {(5, 1, 7), (5, 7, 1)}. We can eliminate (5, 7, 1) by Claim 3. Hence the only case to check is (5, 1, 7). This case does not work. Note: This took 1 check. 4. a = 7. By Claim 2 if b 5 then c 6. Since a , (a, b, c) / {(7, 1, 6), (7, 1, 7)}. Since a , (a, b, c) / {(7, 3, 6), (7, 5, 6)}. Since a , (a, b, c) / {(7, 5, 7), (7, 7, 6), (7, 7, 7)}. The only cases left to check are (a, b, c) {(7, 3, 7), (7, 7, 7)}. By Claim 3 (7, 3, 7) and (7, 7, 7) are eliminated. Note: This took 0 checks. 5. a = 9. By Claim 2 if b 4 then c 5. Since a we have b 5 and c 6. Since a we have (a, b, c) / {(9, 5, 0), (9, 5, 1), (9, 5, 2), (9, 5, 3)}. The only cases left to check are (a, b, c) {(9, 1, 5), (9, 1, 6), (9, 3, 5), (9, 3, 6), (9, 5, 4), (9, 5, 5), (9, 5, 6)}. By Claim 3 (9, 1, 6), (9, 3, 5), (9, 5, 6) are eliminated. Hence the only cases left to check are (a, b, c) {(9, 1, 5), (9, 3, 6), (9, 5, 4), (9, 5, 5)}. None of these work. Note: this took 4 checks. Note: The total number of checks is 24. Question: Is there a proof of Theorem 5.1 with less checks? 8
Math 319 Problem Set #3 Solution 21 February 2002
Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod
More informationEvery Positive Integer is the Sum of Four Squares! (and other exciting problems)
Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Sophex University of Texas at Austin October 18th, 00 Matilde N. Lalín 1. Lagrange s Theorem Theorem 1 Every positive integer
More informationThe Mean Value Theorem
The Mean Value Theorem THEOREM (The Extreme Value Theorem): If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers
More informationk, then n = p2α 1 1 pα k
Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square
More informationThe last three chapters introduced three major proof techniques: direct,
CHAPTER 7 Proving Non-Conditional Statements The last three chapters introduced three major proof techniques: direct, contrapositive and contradiction. These three techniques are used to prove statements
More informationHomework until Test #2
MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such
More informationApplications of Fermat s Little Theorem and Congruences
Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4
More informationIntroduction. Appendix D Mathematical Induction D1
Appendix D Mathematical Induction D D Mathematical Induction Use mathematical induction to prove a formula. Find a sum of powers of integers. Find a formula for a finite sum. Use finite differences to
More informationSome practice problems for midterm 2
Some practice problems for midterm 2 Kiumars Kaveh November 15, 2011 Problem: What is the remainder of 6 2000 when divided by 11? Solution: This is a long-winded way of asking for the value of 6 2000 mod
More information8 Primes and Modular Arithmetic
8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.
More information3 0 + 4 + 3 1 + 1 + 3 9 + 6 + 3 0 + 1 + 3 0 + 1 + 3 2 mod 10 = 4 + 3 + 1 + 27 + 6 + 1 + 1 + 6 mod 10 = 49 mod 10 = 9.
SOLUTIONS TO HOMEWORK 2 - MATH 170, SUMMER SESSION I (2012) (1) (Exercise 11, Page 107) Which of the following is the correct UPC for Progresso minestrone soup? Show why the other numbers are not valid
More informationNIM with Cash. Abstract. loses. This game has been well studied. For example, it is known that for NIM(1, 2, 3; n)
NIM with Cash William Gasarch Univ. of MD at College Park John Purtilo Univ. of MD at College Park Abstract NIM(a 1,..., a k ; n) is a -player game where initially there are n stones on the board and the
More informationVieta s Formulas and the Identity Theorem
Vieta s Formulas and the Identity Theorem This worksheet will work through the material from our class on 3/21/2013 with some examples that should help you with the homework The topic of our discussion
More informationSolution to Homework 2
Solution to Homework 2 Olena Bormashenko September 23, 2011 Section 1.4: 1(a)(b)(i)(k), 4, 5, 14; Section 1.5: 1(a)(b)(c)(d)(e)(n), 2(a)(c), 13, 16, 17, 18, 27 Section 1.4 1. Compute the following, if
More informationHOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!
Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following
More informationThe Prime Numbers. Definition. A prime number is a positive integer with exactly two positive divisors.
The Prime Numbers Before starting our study of primes, we record the following important lemma. Recall that integers a, b are said to be relatively prime if gcd(a, b) = 1. Lemma (Euclid s Lemma). If gcd(a,
More informationElementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.
Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole
More informationRevised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)
Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.
More informationminimal polyonomial Example
Minimal Polynomials Definition Let α be an element in GF(p e ). We call the monic polynomial of smallest degree which has coefficients in GF(p) and α as a root, the minimal polyonomial of α. Example: We
More informationBasic Proof Techniques
Basic Proof Techniques David Ferry dsf43@truman.edu September 13, 010 1 Four Fundamental Proof Techniques When one wishes to prove the statement P Q there are four fundamental approaches. This document
More information3.3 Real Zeros of Polynomials
3.3 Real Zeros of Polynomials 69 3.3 Real Zeros of Polynomials In Section 3., we found that we can use synthetic division to determine if a given real number is a zero of a polynomial function. This section
More information3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
More information2.5 Zeros of a Polynomial Functions
.5 Zeros of a Polynomial Functions Section.5 Notes Page 1 The first rule we will talk about is Descartes Rule of Signs, which can be used to determine the possible times a graph crosses the x-axis and
More informationHomework # 3 Solutions
Homework # 3 Solutions February, 200 Solution (2.3.5). Noting that and ( + 3 x) x 8 = + 3 x) by Equation (2.3.) x 8 x 8 = + 3 8 by Equations (2.3.7) and (2.3.0) =3 x 8 6x2 + x 3 ) = 2 + 6x 2 + x 3 x 8
More information5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
More informationMath 55: Discrete Mathematics
Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 5, due Wednesday, February 22 5.1.4 Let P (n) be the statement that 1 3 + 2 3 + + n 3 = (n(n + 1)/2) 2 for the positive integer n. a) What
More informationNumber Theory. Proof. Suppose otherwise. Then there would be a finite number n of primes, which we may
Number Theory Divisibility and Primes Definition. If a and b are integers and there is some integer c such that a = b c, then we say that b divides a or is a factor or divisor of a and write b a. Definition
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More informationU.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra
U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory
More information15 Prime and Composite Numbers
15 Prime and Composite Numbers Divides, Divisors, Factors, Multiples In section 13, we considered the division algorithm: If a and b are whole numbers with b 0 then there exist unique numbers q and r such
More informationMath 4310 Handout - Quotient Vector Spaces
Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable
More informationDefinition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.
8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent
More informationReal Roots of Univariate Polynomials with Real Coefficients
Real Roots of Univariate Polynomials with Real Coefficients mostly written by Christina Hewitt March 22, 2012 1 Introduction Polynomial equations are used throughout mathematics. When solving polynomials
More informationIterations of sum of powers of digits
International Journal of Computer Discovered Mathematics Volume 0 (2015) No.0 pp.1-4 c IJCDM Received 12 June 2015. Published on-line 12 June 2015 DOI: web site: c The Author(s) 2015. This article is published
More informationWRITING PROOFS. Christopher Heil Georgia Institute of Technology
WRITING PROOFS Christopher Heil Georgia Institute of Technology A theorem is just a statement of fact A proof of the theorem is a logical explanation of why the theorem is true Many theorems have this
More informationSYSTEMS OF PYTHAGOREAN TRIPLES. Acknowledgements. I would like to thank Professor Laura Schueller for advising and guiding me
SYSTEMS OF PYTHAGOREAN TRIPLES CHRISTOPHER TOBIN-CAMPBELL Abstract. This paper explores systems of Pythagorean triples. It describes the generating formulas for primitive Pythagorean triples, determines
More informationSome facts about polynomials modulo m (Full proof of the Fingerprinting Theorem)
Some facts about polynomials modulo m (Full proof of the Fingerprinting Theorem) In order to understand the details of the Fingerprinting Theorem on fingerprints of different texts from Chapter 19 of the
More informationMATH 289 PROBLEM SET 4: NUMBER THEORY
MATH 289 PROBLEM SET 4: NUMBER THEORY 1. The greatest common divisor If d and n are integers, then we say that d divides n if and only if there exists an integer q such that n = qd. Notice that if d divides
More informationWelcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013
Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move
More informationPractice with Proofs
Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using
More information5544 = 2 2772 = 2 2 1386 = 2 2 2 693. Now we have to find a divisor of 693. We can try 3, and 693 = 3 231,and we keep dividing by 3 to get: 1
MATH 13150: Freshman Seminar Unit 8 1. Prime numbers 1.1. Primes. A number bigger than 1 is called prime if its only divisors are 1 and itself. For example, 3 is prime because the only numbers dividing
More informationDomain of a Composition
Domain of a Composition Definition Given the function f and g, the composition of f with g is a function defined as (f g)() f(g()). The domain of f g is the set of all real numbers in the domain of g such
More informationSolutions for Practice problems on proofs
Solutions for Practice problems on proofs Definition: (even) An integer n Z is even if and only if n = 2m for some number m Z. Definition: (odd) An integer n Z is odd if and only if n = 2m + 1 for some
More informationarxiv:0909.4913v2 [math.ho] 4 Nov 2009
IRRATIONALITY FROM THE BOOK STEVEN J. MILLER AND DAVID MONTAGUE arxiv:0909.4913v2 [math.ho] 4 Nov 2009 A right of passage to theoretical mathematics is often a proof of the irrationality of 2, or at least
More informationSettling a Question about Pythagorean Triples
Settling a Question about Pythagorean Triples TOM VERHOEFF Department of Mathematics and Computing Science Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, The Netherlands E-Mail address:
More informationSo let us begin our quest to find the holy grail of real analysis.
1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers
More informationSUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by
SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples
More informationBasic numerical skills: EQUATIONS AND HOW TO SOLVE THEM. x + 5 = 7 2 + 5-2 = 7-2 5 + (2-2) = 7-2 5 = 5. x + 5-5 = 7-5. x + 0 = 20.
Basic numerical skills: EQUATIONS AND HOW TO SOLVE THEM 1. Introduction (really easy) An equation represents the equivalence between two quantities. The two sides of the equation are in balance, and solving
More informationOn the generation of elliptic curves with 16 rational torsion points by Pythagorean triples
On the generation of elliptic curves with 16 rational torsion points by Pythagorean triples Brian Hilley Boston College MT695 Honors Seminar March 3, 2006 1 Introduction 1.1 Mazur s Theorem Let C be a
More informationA Study on the Necessary Conditions for Odd Perfect Numbers
A Study on the Necessary Conditions for Odd Perfect Numbers Ben Stevens U63750064 Abstract A collection of all of the known necessary conditions for an odd perfect number to exist, along with brief descriptions
More informationarxiv:1112.0829v1 [math.pr] 5 Dec 2011
How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly
More informationMath 115 Spring 2011 Written Homework 5 Solutions
. Evaluate each series. a) 4 7 0... 55 Math 5 Spring 0 Written Homework 5 Solutions Solution: We note that the associated sequence, 4, 7, 0,..., 55 appears to be an arithmetic sequence. If the sequence
More information4.3 Lagrange Approximation
206 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION Lagrange Polynomial Approximation 4.3 Lagrange Approximation Interpolation means to estimate a missing function value by taking a weighted average
More informationH/wk 13, Solutions to selected problems
H/wk 13, Solutions to selected problems Ch. 4.1, Problem 5 (a) Find the number of roots of x x in Z 4, Z Z, any integral domain, Z 6. (b) Find a commutative ring in which x x has infinitely many roots.
More information36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?
36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this
More informationPigeonhole Principle Solutions
Pigeonhole Principle Solutions 1. Show that if we take n + 1 numbers from the set {1, 2,..., 2n}, then some pair of numbers will have no factors in common. Solution: Note that consecutive numbers (such
More information8 Divisibility and prime numbers
8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express
More informationMethod To Solve Linear, Polynomial, or Absolute Value Inequalities:
Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column
More informationMATH10040 Chapter 2: Prime and relatively prime numbers
MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us
More informationQuotient Rings and Field Extensions
Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.
More informationTiers, Preference Similarity, and the Limits on Stable Partners
Tiers, Preference Similarity, and the Limits on Stable Partners KANDORI, Michihiro, KOJIMA, Fuhito, and YASUDA, Yosuke February 7, 2010 Preliminary and incomplete. Do not circulate. Abstract We consider
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationSOLVING POLYNOMIAL EQUATIONS
C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra
More informationDigitalCommons@University of Nebraska - Lincoln
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-1-007 Pythagorean Triples Diane Swartzlander University
More informationCHAPTER 5. Number Theory. 1. Integers and Division. Discussion
CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a
More informationChapter 17. Orthogonal Matrices and Symmetries of Space
Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length
More informationMATHEMATICAL INDUCTION. Mathematical Induction. This is a powerful method to prove properties of positive integers.
MATHEMATICAL INDUCTION MIGUEL A LERMA (Last updated: February 8, 003) Mathematical Induction This is a powerful method to prove properties of positive integers Principle of Mathematical Induction Let P
More informationSolutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014
Solutions to Homework 6 Mathematics 503 Foundations of Mathematics Spring 2014 3.4: 1. If m is any integer, then m(m + 1) = m 2 + m is the product of m and its successor. That it to say, m 2 + m is the
More informationGRE Prep: Precalculus
GRE Prep: Precalculus Franklin H.J. Kenter 1 Introduction These are the notes for the Precalculus section for the GRE Prep session held at UCSD in August 2011. These notes are in no way intended to teach
More informationWHERE DOES THE 10% CONDITION COME FROM?
1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay
More informationAs we have seen, there is a close connection between Legendre symbols of the form
Gauss Sums As we have seen, there is a close connection between Legendre symbols of the form 3 and cube roots of unity. Secifically, if is a rimitive cube root of unity, then 2 ± i 3 and hence 2 2 3 In
More informationTAKE-AWAY GAMES. ALLEN J. SCHWENK California Institute of Technology, Pasadena, California INTRODUCTION
TAKE-AWAY GAMES ALLEN J. SCHWENK California Institute of Technology, Pasadena, California L INTRODUCTION Several games of Tf take-away?f have become popular. The purpose of this paper is to determine the
More informationAlgebra Practice Problems for Precalculus and Calculus
Algebra Practice Problems for Precalculus and Calculus Solve the following equations for the unknown x: 1. 5 = 7x 16 2. 2x 3 = 5 x 3. 4. 1 2 (x 3) + x = 17 + 3(4 x) 5 x = 2 x 3 Multiply the indicated polynomials
More informationFive fundamental operations. mathematics: addition, subtraction, multiplication, division, and modular forms
The five fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms UC Berkeley Trinity University March 31, 2008 This talk is about counting, and it s about
More information1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain
Notes on real-closed fields These notes develop the algebraic background needed to understand the model theory of real-closed fields. To understand these notes, a standard graduate course in algebra is
More informationZero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationThe positive minimum degree game on sparse graphs
The positive minimum degree game on sparse graphs József Balogh Department of Mathematical Sciences University of Illinois, USA jobal@math.uiuc.edu András Pluhár Department of Computer Science University
More informationCubes and Cube Roots
CUBES AND CUBE ROOTS 109 Cubes and Cube Roots CHAPTER 7 7.1 Introduction This is a story about one of India s great mathematical geniuses, S. Ramanujan. Once another famous mathematician Prof. G.H. Hardy
More informationSUM OF TWO SQUARES JAHNAVI BHASKAR
SUM OF TWO SQUARES JAHNAVI BHASKAR Abstract. I will investigate which numbers can be written as the sum of two squares and in how many ways, providing enough basic number theory so even the unacquainted
More informationInteger roots of quadratic and cubic polynomials with integer coefficients
Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street
More informationOn the largest prime factor of x 2 1
On the largest prime factor of x 2 1 Florian Luca and Filip Najman Abstract In this paper, we find all integers x such that x 2 1 has only prime factors smaller than 100. This gives some interesting numerical
More informationLinear Algebra Notes
Linear Algebra Notes Chapter 19 KERNEL AND IMAGE OF A MATRIX Take an n m matrix a 11 a 12 a 1m a 21 a 22 a 2m a n1 a n2 a nm and think of it as a function A : R m R n The kernel of A is defined as Note
More informationMATH 22. THE FUNDAMENTAL THEOREM of ARITHMETIC. Lecture R: 10/30/2003
MATH 22 Lecture R: 10/30/2003 THE FUNDAMENTAL THEOREM of ARITHMETIC You must remember this, A kiss is still a kiss, A sigh is just a sigh; The fundamental things apply, As time goes by. Herman Hupfeld
More informationComputing exponents modulo a number: Repeated squaring
Computing exponents modulo a number: Repeated squaring How do you compute (1415) 13 mod 2537 = 2182 using just a calculator? Or how do you check that 2 340 mod 341 = 1? You can do this using the method
More informationAlgebraic and Transcendental Numbers
Pondicherry University July 2000 Algebraic and Transcendental Numbers Stéphane Fischler This text is meant to be an introduction to algebraic and transcendental numbers. For a detailed (though elementary)
More informationSolving Linear Systems, Continued and The Inverse of a Matrix
, Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing
More informationOverview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model
Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written
More informationMA107 Precalculus Algebra Exam 2 Review Solutions
MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write
More informationZeros of a Polynomial Function
Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we
More informationCubic Functions: Global Analysis
Chapter 14 Cubic Functions: Global Analysis The Essential Question, 231 Concavity-sign, 232 Slope-sign, 234 Extremum, 235 Height-sign, 236 0-Concavity Location, 237 0-Slope Location, 239 Extremum Location,
More informationCartesian Products and Relations
Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special
More informationFACTORING AFTER DEDEKIND
FACTORING AFTER DEDEKIND KEITH CONRAD Let K be a number field and p be a prime number. When we factor (p) = po K into prime ideals, say (p) = p e 1 1 peg g, we refer to the data of the e i s, the exponents
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationNormal distribution. ) 2 /2σ. 2π σ
Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a
More informationThe Taxman Game. Robert K. Moniot September 5, 2003
The Taxman Game Robert K. Moniot September 5, 2003 1 Introduction Want to know how to beat the taxman? Legally, that is? Read on, and we will explore this cute little mathematical game. The taxman game
More informationInteger Factorization using the Quadratic Sieve
Integer Factorization using the Quadratic Sieve Chad Seibert* Division of Science and Mathematics University of Minnesota, Morris Morris, MN 56567 seib0060@morris.umn.edu March 16, 2011 Abstract We give
More informationIntermediate Value Theorem, Rolle s Theorem and Mean Value Theorem
Intermediate Value Theorem, Rolle s Theorem and Mean Value Theorem February 21, 214 In many problems, you are asked to show that something exists, but are not required to give a specific example or formula
More information